\begin{question} What repeated applying of $\Phi_{\ast}$ and $\Phi^{\ast}$ to "other" leads to? Particularly, does repeated applying $\Phi_{\ast}$ and/or $\Phi^{\ast}$ to the node "other" lead to finite or infinite sets? \end{question}

See \href [Algebraic General Topology]{http://www.mathematics21.org/algebraic-general-topology.html} for definitions of used concepts.

The known part of the diagram is considered in \href[this file]{http://www.mathematics21.org/binaries/addons.pdf}.

% You may use many features of TeX, such as % arbitrary math (between $...$ and $$...$$) % \begin{theorem}...\end{theorem} environment, also works for question, problem, conjecture, ... % % Our special features: % Links to wikipedia: \Def {mathematics} or \Def[coloring]{Graph_coloring} % General web links: \href [The On-Line Encyclopedia of Integer Sequences]{http://www.research.att.com/~njas/sequences/}